(reconsidered; no time at the moment to bring post into satisfactory form)

]]>**If $\mathcal{D}$ is a category, one may define a category with the same objects but having morphisms precisely all possible families of parallel morphisms of $\mathcal{D}$, in the obvious way.**

**It appears practically certain that this is a standard construction with a usual technical name; would you please tell what it is?**

**In short: it is possible to define compositions of one hom-set with another hom-set, and thus get categories having the hom-sets themselves as the morphisms; is there a usual technical term for this?**

(In a sense, it is the straightforward generalization of what are called “Minkowski sums” or “sum-sets” in the special situation of commutative monoids, but this question concerns the general construction for any category. It can probably also made into an endofunctor of the category of all small categories.)

Perhaps needlessly, details:

Suppose $\mathcal{D}$ is a category.

Let $\mathcal{D}^+$ denote the category which has

- precisely the same objects as $\mathcal{D}$
for arbitrary objects $O,O'$ of $\mathcal{D}^+$ we define the hom-‘set’ to be

$\mathcal{D}^+(O,O') :=$ class of all class-indexed families $\mathcal{M}=\{ h_i\colon i\in I\}$,

with $I$ a class and each $h_i\in\mathcal{D}(O,O')$.

(We note that each morphism is a class of parallel morphisms.)

Composition is defined in the obvious way: if $O,O',O''$ are objects,

and if $\{ h_i\colon i\in I\}\in\mathcal{D}^+(O,O')$ and $\{ h_i'\colon i\in I'\}\in\mathcal{D}^+(O',O'')$, then

$\{ h_i'\colon i\in I'\}\circ\{ h_j\colon j\in I\} := \{ h_i'\circ h_j\colon (i,j)\in I'\times I\}$

(We note that $\mathrm{dom}(h_i')=\mathrm{cod}(h_j)$ for all indices, so all the compositions are defined.)

(We note that the identity morphism at an object of $\mathcal{D}^+$ is the singleton-indexed class containing only the singleton-morphism of $\mathcal{D}$ at that object.)

]]>**How would you call the full subcategory of a functor category [J,C] consisting only of those functors which have a limit in C ?**

Of course, the description just given seems to do just fine, but do you prefer a technical term for that?

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